https://artofproblemsolving.com/wiki/index.php?title=Base_(topology)&feed=atom&action=history Base (topology) - Revision history 2022-01-20T04:46:19Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Base_(topology)&diff=33888&oldid=prev Azjps: stub 2010-03-17T02:01:01Z <p>stub</p> <p><b>New page</b></p><div>A '''basis''' or '''base''' of a [[topology]] is a collection &lt;math&gt;\mathcal{B}&lt;/math&gt; of sets in a topological space &lt;math&gt;(X,\tau)&lt;/math&gt; that classify the set of [[open set]]s of the space. <br /> <br /> #For any basis &lt;math&gt;\mathcal{B}&lt;/math&gt;, the union of the sets in &lt;math&gt;\mathcal{B}&lt;/math&gt; is equal to &lt;math&gt;X&lt;/math&gt;. Phrased differently, for any element &lt;math&gt;x \in X&lt;/math&gt;, there exists a basis set &lt;math&gt;A \in \mathcal{B}&lt;/math&gt; such that &lt;math&gt;x \in A&lt;/math&gt;. <br /> #For any two sets &lt;math&gt;A, B \in \mathcal{B}&lt;/math&gt;, given an element &lt;math&gt;x \in A \cap B&lt;/math&gt;, then there exists another set &lt;math&gt;C \in \mathcal{B}&lt;/math&gt; such that &lt;math&gt;x \in C \subset A \cap B&lt;/math&gt;. <br /> <br /> This definition is very useful for comparing different topologies. In particular, we have the following theorem:<br /> <br /> ''Theorem'': In a space &lt;math&gt;X&lt;/math&gt;, given two topologies &lt;math&gt;\tau&lt;/math&gt; and &lt;math&gt;\tau'&lt;/math&gt;, then &lt;math&gt;\tau \subset \tau'&lt;/math&gt; [[iff]] for any basis element &lt;math&gt;B \in \mathcal{B}&lt;/math&gt; and any element &lt;math&gt;x \in B&lt;/math&gt;, there exists a basis element &lt;math&gt;B' \in \mathcal{B}'&lt;/math&gt; such that &lt;math&gt;x \in B' \subset B&lt;/math&gt;. <br /> <br /> == Sub-basis ==<br /> A '''sub-basis''' &lt;math&gt;\mathcal{S}&lt;/math&gt; of &lt;math&gt;X&lt;/math&gt; is a collection of sets whose union is &lt;math&gt;X&lt;/math&gt;. The collection of intersection of sets in &lt;math&gt;\mathcal{S}&lt;/math&gt; forms a basis &lt;math&gt;\mathcal{B}&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt;. <br /> <br /> {{stub}}<br /> <br /> [[Category:Topology]]</div> Azjps